1. There are n people in a town, having the length of their name greater than 3. What is the number of people in the town such that you are guaranteed to find at least two people with the same second alphabet in their name?

• 26

• 54

• 52

• 27

Correct Answer :27

2. Given a set of integers from 51 to 250, { 51, 52, 53,......,250}. What is the least number of integers that can be selected such that any two of the chosen integers are consecutive?

• 50

• 101

• 51

• 100

Correct Answer :101

3. Which of the following statements is FALSE?

• • P(n) =n ³ -n is divisible by 3, ∀ n ∈ Z

  • n ² > 2n +1 ∀ n Z ⁺

  • P(i):1+3+5+.........+2i−1=i ²

  • n<2 ⁿ ∀ n ∈ Z ⁺

Correct Answer :-n ² > 2n +1 ∀ n Z ⁺

4. Is the given inequality true?

2 ⁿ <n!∀n∈N

• Yes

• No

Correct Answer :- No

5. A black-coloured (opaque) bag contains 15 red gloves and 15 green gloves, Krishna takes out one glove at random each time. How many gloves must he take out to be sure that he has at least 2 red gloves?

• 17

• 15

• 30

• 16

Correct Answer :17

6. How many of the following statements are true?

I. If we pick 9 cards from a deck of 52 cards, we are guaranteed that there is a suit with at least 3 cards.

II. If you pick four numbers from 1 to 8 (including 1 and 8), then some two of them will add up to 9.

III. Every 2 ⁿ x 2 ⁿ checkerboard board with one square removed can be tiled using triomino.

• • 0

  • 1

  • 2

  • 3

Correct Answer : 2

7. For positive integer n, 10 ^n-2 > 81n, if

• n>5

• n<4

• n≤4

• n≥5

Correct Answer :-n≥5

8. A basket containing 60 walnuts was kept in the veranda, and 10 squirrels randomly came and took away some walnuts. After some time the gardener found the basket empty. Which of the following conclusions is always true for the above-given situation?

• At least one squirrel went back with at most 6 walnuts.

• At most one squirrel went back with at least 6 walnuts.

• At least one squirrel went back with at least 6 walnuts.

• At most one squirrel went back with at most 6 walnuts.

Correct Answer :At least one squirrel went back with at least 6 walnuts.

9. State whether the following statement is True/False:

For all n∈N, 6 ⁿ -1 is a multiple of 5.

• True

• False

Correct Answer :True

10. Madhu, Anushka, and Prasad bought a White Chocolate consisting of 𝑛 blocks, the chocolate is distributed equally among all the three members. Madhu likes to enjoy the chocolate, having it one block at a time. How many total breaks does Madhu require to finish her share of chocolate?

• n−1

• n/3

• n/4

• n/3-1

Correct Answer :- n/3-1

WEEK-6 DETAILED SOLUTION